We want to select a minimum cost tree $S=(P,T)$ such that $P$ contains exactly one node from each of the $m$ clusters. We use undirected edge variables $x_{ij} \; \forall \{i,j\} \in E$ and node variables $z_v \; \forall v \in V$ for an instance graph $G=(V,E)$ and edge weights $w_{ij}$.  

The objective function is
\begin{align}
\min \quad \sum_{e=\{i,j\} \in E} x_{ij}w_{ij} & \\
\sum_{i \in V_k} z_{i} &= 1 & \forall k=1,...,m \label{eq:egmstcluster} \\
\sum_{\{i,j\} \in E(S)} x_{ij} & \leq \sum_{v \in S \setminus \{k\}} z_v & \forall S \subseteq V, |S|\geq 2, \forall k \in S \label{eq:egmstsub} \\
\sum_{\{ij\} \in E} x_{ij} & \geq \sum_{i \in V} z_i - 1 \label{eq:egmsttree} \\
\end{align}

and the variables:
\begin{align}
x_{ij} & \in \{ 0,1 \} & \forall \{i,j\} \in E \\
z_i & \in \{ 0,1 \} & \forall i \in V
\end{align}

Constraint~\eqref{eq:egmstcluster} defines that we need to select one node per cluster.
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With equation~\eqref{eq:egmstsub} we ensure that our subgraph has no cycles and that there is no selected edge which does not link 2 selected nodes, because otherwise we would have for $S=\{ i,j\}: \sum_{\{i,j\} \in E(S)} x_{ij} = 1 \nleq 0 = \sum_{v \in S \setminus \{j\}}z_v$ and this violates the constraint.
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Lastly constraint~\eqref{eq:egmsttree} ensures that our selected solution is a tree.